Lecture 11: pn junctions under bias
Contents
1 Introduction
1
2 Forward bias
2.1 Carrier injection . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
3 Forward bias current
3.1 Band gap dependence . . . . . . . . . . . . . . . . . . . . . . .
5
7
4 Reverse bias
8
1
Introduction
A pn junction at equilibrium is characterized by a depletion region where
there are no charge carriers (except for those created and annihilated dynamically) and a contact potential. The contact potential is related to the
dopant concentration in the p and n sides with higher concentrations leading
to larger contact potentials. This, in turn, is related to the position of the
Fermi levels in the p and n sides since a higher dopant concentration pushes
the Fermi level closer to the valence or conduction band. The pn junction
can be biased by connecting to an external circuit and there are two types
of biasing (similar to the arguments for the metal-semiconductor Schottky
junction)
1. Forward bias
2. Reverse bias
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Figure 1: pn junction under (a) equilibrium, (b) forward and (c) reverse bias.
The depletion width shrinks in forward bias and expands in reverse bias.
Adapted from Semiconductor device physics and design - Umesh Mishra and
Jasprit Singh.
2
Forward bias
Consider a pn junction under forward bias. This is achieved by connecting
the p side to the positive terminal of an external power source and the n side
to the negative terminal. In reverse bias, the connections are interchanged.
Equilibrium, forward, and reverse bias connections are shown in figure 1. In
the forward bias the external potential (V ) opposes the contact potential,
V0 , that develops in equilibrium. The effect of this is that the net potential
at the junction is lowered. In the presence of an external potential the Fermi
levels no longer line up but are shifted. This shift can be seen in the band
diagram, summarized in figure 2. The application of the external potential,
in forward bias, shifts the n side up with respect to the p side, see figure 2.
This leads to a lowering of the barrier for injection of electrons from the n to
the p side (there is a similar lowering of the barrier for holes to be injected
from p to n side) and leads to a current in the circuit. This current is due to
the injection of minority carriers in the pn junction.
2.1
Carrier injection
Consider a forward biased pn junction showing the change in concentration
of carriers along the length of the junction, moving from the p to the n side,
figure 3. In equilibrium, the carrier concentrations in the p and n sides are
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Figure 2: Band diagram of pn junction under (a) equilibrium and (b) forward
bias. While Fermi levels line up in equilibrium in the presence of an external
potential the levels shift by an amount proportional to the applied potential.
Adapted from Semiconductor device physics and design - Umesh Mishra and
Jasprit Singh.
Figure 3: Current in a pn junction is due to injection of minority carriers in
forward bias. These excess carriers can diffuse before recombining with the
majority carriers. Adapted from Principles of Electronic Materials - S.O.
Kasap.
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given by
n2i
NA
n2
= i
ND
pp0 = NA ; np0 =
nn0 = ND ; pn0
(1)
Now, extra carriers are injected due to the forward bias, as shown in figure
3. These extra carriers are minority carriers and diffuse some distance before
recombining. These minority carriers constitute the current in a forward
biased pn junction since they are constantly being supplied by the external
potential . In figure 3, pn (0) and np (0) represent the carriers that are injected
due to the applied forward bias. Their concentration is related to the reduced
barrier for carrier injection
e(V0 − Vext )
)
kB T
e(V0 − Vext )
)
exp(−
kB T
pn (0) = pp0 exp(−
np (0) = nn0
(2)
where V0 is the barrier potential and Vext is the external potential during
forward bias and the equilibrium concentrations are defined in equation 1.
These are minority carriers and diffuse a short distance before getting annihilated. This can be seen in the concentration plot in figure 3 where the
initial high concentration at the depletion region interface, pn (0) and np (0),
gets reduced to the equilibrium concentration, as we move deeper into the
bulk of the n and p regions respectively.
The distance traveled by the minority carriers before recombination is called
the minority carrier diffusion length (Lh or Le ). Using the usual formulation for one dimensional diffusion this can be written in terms of a diffusion
coefficient (Dh or De ) and carrier lifetime (τh or τe ).
Lh =
p
p
Dh τh ; Le =
De τe
(3)
The diffusion coefficient is related to the electron and hole mobility values
(µh and µe ) by the Einstein relation.
Dh =
kB T µh
kB T µe
; De =
e
e
(4)
Consider some typical values for Si. We can the take electron and hole mobilities for undoped Si, where µe is 1350 cm2 V −1 s−1 and µh is 450 cm2 V −1 s−1 .
Using equation 4 the electron and hole diffusivities are calculated to be 34.93
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cm2 s−1 and 11.64 cm2 s−1 respectively. Typical values for the carrier lifetimes
(time before the electron or hole recombines and gets annihilated) are of the
order of ns. This is different from the carrier scattering time which is of the
order of ps (10−12 s) and is defined as the time between two successive collisions. A carrier can undergo multiple collisions before recombining. Taking
a carrier recombination time of 1 ns the diffusion lengths can be calculated
using equation 3. This gives an electron diffusion length of 1.9 µm and hole
diffusion length of 1.08 µm. Thus, typical diffusion lengths in a pn junction,
where the injected minority carriers recombine, is of the order of µm.
3
Forward bias current
Consider the pn junction schematic shown in figure 3. The excess electron
concentration at the interface between the depletion width and the p side
is np (0) and similarly the excess hole concentration on the n side is pn (0).
These values are given by equation 2. These excess carriers are replenished
by the applied voltage of the external circuit so that a current flows through
the entire circuit.
Consider the n side of the junction, where the excess minority carriers are
holes. When the length of the n region is longer than the diffusion length,
the hole concentration at a distance x from the depletion region, marked in
figure 3, is given by
x
)
Lh
x
= ∆pn (0) exp(− )
Lh
pn (x) = pn (0) exp(−
excess holes ⇒ ∆pn (x) = pn (x) − pn0
(5)
The excess holes is above the base hole concentration in the n side, which
is very small. The hole diffusion current (JD, hole ) is then defined by the
diffusion coefficient and concentration gradient
JD, hole = −eDh
dpn (x)
d∆pn (x)
= −eDh
dx
dx
(6)
This is similar to Fick’s first law of diffusion. Substituting for pn (x) using
equation 5 and evaluating, the hole current is given by
JD, hole =
eDh
x
∆pn (0) exp(− )
Lh
Lh
(7)
Similarly, there will be a current due to electron diffusion in the p region,
which can be written similar to equation 7. The total diffusion current, which
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Figure 4: Total diffusion current in a pn junction. This is the sum of the
electron and hole current and also a drift component due to the electric field.
The diffusion current is due to the injection of minority carriers. Adapted
from Principles of Electronic Materials - S.O. Kasap.
is the sum of the electron and hole current, is a constant and independent of
position. This is shown schematically in figure 4.
Thus, the total diffusion current, due to electron and holes, can be evaluated
at x = 0. The hole diffusion current at this position can be written using
equations 7 and 2 and the law of mass action.
eDh
eDh
JD, hole =
∆pn (0) =
(pn (0) − pn0 )
Lh
Lh
eDh pn0
eV
JD, hole =
[exp(
) − 1]
(8)
Lh
kB T
eV
eDh n2i
[exp(
) − 1]
JD, hole =
Lh ND
kB T
It is possible to write a similar expression for the current due to the diffusion
of electrons. This can be written as
eDe n2i
eV
JD, electron =
[exp(
) − 1]
(9)
Lh NA
kB T
Thus, the total diffusion current, due to both electrons and holes is given by
the sum of equations 8 and 9.
JD = (
eV
Dh
De
+
)en2i [exp(
) − 1]
Lh ND
Lh NA
kB T
6
(10)
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This is called the Schockley equation and gives the total current due to
diffusion in the forward bias and its dependence on the applied voltage. The
first part of equation 10 is called the reverse saturation current density(Js0 ).
De
Dh
+
)
Lh ND
Lh NA
eV
= Js0 [exp(
) − 1]
kB T
Js0 = e n2i (
JD
(11)
This forward bias current is due to the diffusion of the minority carriers in
the pn junction. Lh and Le are the diffusion lengths of the minority carriers
and they are typically smaller than the dimensions of the p and n regions.
This is called a long diode. If the diode dimensions are smaller than the
diffusion lengths, it is called a short diode, and Lh and Le are replaced by le
and lh , the diode dimensions.
Some of the minority carriers diffusing across the junction will recombine in
the depletion region. These are also replenished by the electrons and holes
supplied by the external circuit. This current is called the recombination
current and is also exponentially dependent on the applied voltage. Combining both terms (diffusion and recombination current) the total current in
a forward biased pn junction is given by
J = J0 exp(
eV
)
ηkB T
(12)
where J0 is a new constant and η is called an ideality factor , with a value
between 1 and 2. When η is close to 1 the current is mostly diffusion current
and when η is close to 2 the current is mostly due to minority recombination.
The effect of recombination is to lower the overall current in the pn junction.
The I-V characteristics in forward bias for different semiconductors is shown
as a semilog plot in figure 5. Since the current depends exponentially on
the applied voltage, the semilog plot is a straight line with different slopes
depending on the semiconductor and value of η.
3.1
Band gap dependence
Consider the reverse saturation current (Js0 ) shown in equation 11. This
contains the term n2i , which is the intrinsic carrier concentration. This is a
material property, for a given temperature, and depends on the band gap.
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Figure 5: Forward bias I-V plot for a pn junction. Adapted from Principles
of Electronic Materials - S.O. Kasap.
This can be incorporated in the expression for Js0 .
eV
eV
) − 1] ≈ Js0 exp(
)
kB T
kB T
Dh
De
Js0 = e n2i (
+
)
Lh ND
Lh NA
eVg
n2i = Nc Nv exp(−
)
kB T
De
e(V − Vg )
]
+
) e (Nc Nv ) exp[
Lh NA
kB T
e(V − Vg )
JD = J1 exp[
]
kB T
JD = Js0 [exp(
JD = (
Dh
Lh ND
(13)
Vg here represents the band gap Eg converted into a potential (dividing by
e). The diffusion current then depends on a temperature dependent constant
multiplied by a term that depends on the band gap. The forward bias I-V
characteristics for different semiconductors are plotted in figure 6. For a
given current value, the voltage required is higher with higher Vg (higher
band gap).
4
Reverse bias
In forward bias the current increases exponentially with the applied voltage.
The external potential opposes the in-built potential and has the effect of
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I−V plots for different semiconductors
Current density (Acm−2)
0.8
0.7
0.6
GaAs
Si
Ge
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
Voltage (V)
0.8
1
Figure 6: Forward bias I-V plots for pn junctions of three different semiconductors. The plots were generated in MATLAB for the same value of donor
and acceptor concentrations. The band gap of Ge is 0.66 eV , Si is 1.1 eV
and GaAS is 1.43 eV . For a certain current density, shown by dotted lines,
the applied voltage required increases with increasing band gap.
lowering the barrier for the electrons and holes. In reverse bias, the applied
external potential is in the same direction as the contact potential. This
is shown schematically in figure 7. The p side is connected to the negative
potential and the n side is connected to the positive potential. The effect of
the reverse bias on depletion width is shown schematically in figure 1. The
reverse bias causes the depletion region width to increase since the majority
carriers are attracted to the external potential. In the energy band diagram
the Fermi levels are shifted, as shown in figure 8. This is opposite to the
direction of forward bias, shown in figure 2. Because of the higher barrier,
diffusion current is negligible in reverse bias. There is however a small current
that flows through the pn junction, called the reverse saturation current.
This current is a constant (independent of reverse bias voltage) and is generated by drift of the thermally generated carriers in the depletion region.
Electron and holes dynamically generated in the depletion region get accelerated towards the n and p side due to the applied voltage and this leads to
a small reverse saturation current, also called drift current. This is given
by Js0 , shown in equation 11.
Js0 = e n2i (
De
Dh
+
)
Lh ND
Lh NA
(14)
This is typically orders of magnitude smaller than the forward bias current.
The I-V characteristics of a pn junction, for both forward and reverse bias, is
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Figure 7: Reverse bias configuration for a pn junction. The total potential
at the depletion region is increases. This has the effect of increasing the
depletion width making it harder for carriers to cross the junction. Adapted
from Principles of Electronic Materials - S.O. Kasap.
Figure 8: Band diagram of pn junction under reverse bias. The Fermi level on
the n side shifts down leading to a overall increase in the junction potential.
Adapted from Semiconductor device physics and design - Umesh Mishra and
Jasprit Singh.
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Figure 9: I-V characteristics of a pn junction. The diode symbol is shown
in the inset. There is an exponentially increasing current in forward bias
while the reverse bias current is orders of magnitude smaller. Adapted from
Semiconductor device physics and design - Umesh Mishra and Jasprit Singh.
shown schematically in figure 9. The forward bias current is orders of magnitude higher than the reverse bias current so that a pn junction acts as a
rectifier. It conducts only in one direction (forward) and does not conduct
in the other direction (reverse). The energy band information is summarized
in figure 10.
Consider an example of Si pn junction, operating at room temperature.
The acceptor concentration is 1016 cm−3 and the donor concentration is
1015 cm−3 , on the p and n side respectively. While the carrier mobility
normally decreases with doping, we can use the mobility values for intrinsic
Si, i.e. µe = 1350 cm2 V −1 s−1 and µh = 450 cm2 V −1 s−1 . The minority
carrier lifetime is typically of the order of ns, we can use τe = 50 ns and
τh = 100 ns. Since this is Si at room temperature, ni = 1010 cm−3 , the
intrinsic carrier concentration. Using equations 3 and 4 it is possible to calculate the carrier diffusion lengths and diffusivites. Substituting these values
in equation 11, it is possible to calculate the reverse saturation current density to be 2.15 × 10−10 Acm−2 . This is a very small current that is present in
reverse bias. The forward bias current increases exponentially with applied
voltage. The exponential dependence can be clearly seen in a semilog plot
and it is plotted along with the overall I-V behavior in figure 11.
Both the pn and Schottky junction show rectification behavior on appli11
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Figure 10: Energy band diagram of a pn junction under (a) equilibrium,
(b) forward bias, and (c) reverse bias. (d) The reverse saturation current
due to thermal generation of carriers is shown. Adapted from Principles of
Electronic Materials - S.O. Kasap.
Forward bias log plot
0
I−V plot
0.15
(a)
Current density (Acm−2)
Current density (Acm−2)
10
−2
10
−4
10
−6
10
−8
10
0
0.12
0.09
0.06
0.03
0
−10
10
(b)
0.2
0.4
Voltage (V)
0.6
−0.4 −0.2
0
0.2
0.4
Voltage (V)
0.6
Figure 11: (a) Semilog plot of forward bias current vs. voltage for Si pn
junction. The exponential increase in current with voltage is seen in the
straight line plot. (b) I-V characteristics of the pn junction. The forward
bias current is orders of magnitude higher than the reverse bias current. The
plots were generated in MATLAB.
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cation of an external potential. In both cases the behavior arises from the
lowering of the potential barrier on application of forward bias and the current increases exponentially with voltage. For these junctions, it is possible to
define a rectification ratio, which is the ratio of the forward bias to reverse
bias current, for the same absolute voltage value. pn junctions have a typical
rectification ratios of 107 − 1010 , while Schottky junctions have values around
103 − 106 , making pn junctions a better rectifier than metal-semiconductor
Schottky junctions.
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